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(Nonetheless there exists a motivation for belief in their existence: as in the case of photons interacting with nucleons—where the strongly interacting vector mesons seem to dominate-in the case of neutrino interactions—as was pointed out by Adler -7-mesons should be important). In other words, in all the processes we are going to discuss, strongly interacting particles appear (as real or virtual particles) which may produce shadowing effects.
So, the crucial point in our discussion is an understanding of a multiple scattering process of strongly interacting particles inside of nuclear matter (or more generally: just a multiple scattering process with forces strong enough to insure the existence of multiple scattering). Hence we shall start with the very successful model of such processes: the Glauber model.
2. Description of Multiple Scattering
2.1. General Remarks
To construct the relevant formulae for the theory of multiple scattering one can employ various models of potential scattering. First let me quote the well-known formulae: one particle scatters from a collection of A particles at very small angles (in the Glauber model ).
This is to a very good approximation a two dimensional process. The individual amplitude
where k is the momentum of the incident particle in laboratory frame
A is the two-dimensional momentum transfer b is the impact parameter xi(b) is the phase shift which characterizes the incident particle--jth nucleon elastic scattering amplitude.
is called the profile of the jth nucleon, incident particle collision. Assuming
x(b) = Ï xi(b-8;)
and assuming that the particle goes through the target so fast that all the nucleons are 'frozen’ at certain positions, we get for the amplitude
where Vi and V, are the initial and final wave functions of the target nucleus.
One can produce many arguments which make this important formula plausible. One can use, e.g., an optical description of attenuation of a wave penetrating a medium. One can also use some arguments based on approximate solutions of the wave equation of the incident particle interacting through potentials with the target particles.
For instance, in the case of the Schrödinger equation
in the limit E+00, and for the incident particle moving along the z axis, we present the solution in the form
**(x, y, z) = eiks p(x, y, z).
If the potential is smooth enough (so that second derivatives of y can be neglected), one can show that y satisfies the approximate equation
2 Notice that to have scattering in the limit Eno we have to have V~EV' where V' is energy independent. Otherwise the high energy solution of the Schrödinger equation reduces to the Born approximation.
hence, neglecting second derivatives of y, we obtain the following equation for :
we recover additivity of phase shifts.
There are many simplifications made in obtaining the fundamental formula (2.1); the reliability of this formula is of primary importance. The most complete analysis one can perform is presumably to employ the Watson multiple scattering theory, but we shall not present it here.
In fact it is amazing that (2.1) works so well. Even in the conceptually simplest cases of relativistic potential scattering one can give examples in which it breaks down.
Examples Example 1. Dirac particle with anomalous magnetic moment in a given electromagnetic static field (notation from Bjorken and Drell ):
This equation was worked out in ref. .
Y2K0Fw= -2Y2K (001Ez +002Ey +003E2)
+212K (023B, +031B, +012B.)
This is because the right-hand side of (2.3) does not contain the energy, E.
We multiply eq (2.3) from the left by 12(1+az) and get (note that (1+az) (1 - a3) = azo=0)
where the transverse components (in x, y plane) are marked 1. Thus, finally, we find
So, if the anamalous magnetic moment K =0, we end up with an expression which is virtually the same as in the case of the Schrödinger equation:
ū(k”) you (k)i s :b exp (ia.b) [1–exp(- ie
exp(-ie S ** dz V(b, 2)) ]
So, in this case we also have additivity of phase shifts—hence the Glauber model: But when K+0 the principle of additivity of phase shifts breaks down. Let us consider this case in more detail.
a From the equation -2 +eV – KB (2.•B1-ia.•E) =0 we can eliminate the 'trivial
dependence on V by substituting
dz'V (xy, z)
R=Fexp(-ie | «V (2, 4, =)),
F is a four spinor but we can reduce it to an equation for a two component spinor because F has to satisfy the relation
which is in fact a system of first order differential equations for two unknown functions (the two components of the spinor x). Call
A (x, y, z) =K (01.B.-10.• E10,).
[Q (x, y, z), A (x, y, z')]+0.