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where 8 is a small complex number, 181–10-3«1, which gives a measure of CP nonconservation. | Kjo) and | K29) are eigenstates of CP. Indeed

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We observe these two different decays by looking at decaying K mesons in the beam.

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So, after some distance only the K2 beam is in existence. When we let the beam hit another target we can regenerate K, mesons because Ko and Ko interact differently with matter, they are absorbed differently. (e.g., K°(S=-1)+p—4°(S=-1)+7+ while Ko(S=+1) cannot produce Ao). In figure 18 the so-called transmission regeneration is sketched.

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One can, however, also observe diffractive production of K,o from K 1° on individual nuclei. Cross sections for such diffractive coherent production processes were recently measured for copper and lead nuclei [26]. In order to measure these cross sections, one has to get off the forward direction where the transmission regeneration (which comes from a coherent process whose coherence extends over the whole block of matter) dominates. The amplitude (neglecting 8) is

MKL-K,o=12(Ko | T | K°)-12(Ko | T |ð).

Hence MKL --K.• is given by the elastic scattering amplitudes of Ko and Ko from the nucleus

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(Ko | T |ð)=M5°(42) = 2 / db exp (ia.b) {1– exp [ixx•(6)]},




where, as was already shown for copper and lead target nuclei, it is enough to take the large-A approximation:

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The elementary amplitudes can be gotten from K+-nucleon scattering amplitudes assuming isospin symmetry:

fxon (0) = f**p(0), fron(0) =fk-p(0),

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(i) the imaginary parts are obtained from the optical theorem, e.g.


k. Im fk+p=


(ii) the real parts from dispersion relations (for more details compare [26]). The amplitudes

for neutrons are then obtained using some further acrobatics, as referred to in [26]. In any case, our knowledge of these amplitudes is rather poor.

The real and imaginary parts of the elastic Ko and Ko-nuclear amplitudes are sketched below. The uncertainties of our knowledge of these amplitudes are also shown [26].

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From this picture it is clear that neutrons are much more effective in regeneration than protons (the difference in absorption of K and Ko is much bigger in the case of neutrons).

One can get excellent fits to the differential cross sections dok L-K3/dA? by making the neutron and proton distributions different. One gets the following nuclear parameters from the best fits [26].

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1. Assuming that we can trust the input data (structure of K,' and KL', skon, from, fxop, skop amplitudes) we obtained a very important piece of information about the target nuclei: the neutron distribution. This is so because neutrons are more effective in regeneration than protons. In this case we have not, however, obtained any new information about the elementary processes and the structure of K,' and KL.

2. The main feature of the K regeneration process seems to be very general, however: The process of diffractive production consists in rearrangement of the "components” (understood in a very broad sense) of the incident particle (system): but components undergo only elastic scattering. This description is common to many models of diffractive dissociation (and excitation).

3. In describing the regeneration process KVK,' we drew heavily on the known structure of K2' and K,': they are superpositions of Ko and Ko, whose elastic scattering from nucleons is reasonably well known.


4.1. Generalizing to Other Diffractive Production (and Excitation) Processes

First of all, the components of the incident and the produced states are, in general, not degenerate: their invariant masses differ. This fact may introduce some important corrections at low energies. But in the limit of very high energies and small momentum transfer, all such effects disappear. Let us take, e.g., two such states and give them the same momentum p. Then their energies differ:

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(The only important thing in these approximations is to have very large longitudinal momenta in the initial and final states.) As long as the time of the passage through (or the interaction with) the target is




we can consider the states to be degenerate because their relative phase factor during the collision is very small and we have exp[-i(E-E*)]1 to very good accuracy.

In fact the same argument shows that the incident state and the produced state can also be considered degenerate in the limit p-(the fact that p will also change slightly during the collision does not change this conclusion). Note that the degeneracy appears in the laboratory system, where the incident and produced systems move fast.

One can also show that in this limit (p=00) the longitudinal momentum transfer can also be neglected. So, in the limit p=0 all the states taking part in diffractive production processes can be considered degenerate and the longitudinal momentum transfer neglected. As was said, however, for low p's some important corrections may appear.

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Summing up:

Our procedure for evaluating diffractive production in the limit of very high energy consists of two steps: (i) Find a "plausible” model and identify the components of the incident and outgoing

systems. (ii) Compute the transition matrix element by making the components scatter elastically

from the target.

If one can implement such a program, one can treat diffractive production processes on nucleons (elementary targets) and nuclei (composite targets) on the same footing: the only difference is that the components scatter elastically from a nucleon in the first and from a nucleus in the second case.

We shall start discussing the approach outlined at the beginning of section 4 with restrictions (4.3) because this scheme contains a large class of known models of diffractive production, including diffractive processes in QED [21], [22], [27], [28].

] Without going into any details let me sketch an example of such an approach in the case of the process of proton dissociation p->n+*+. As in QED (compare e.g., lectures by W. Czyż in ref. [S4]) we express the states of the physical proton and the physical neutron-pion pair through the "bare" states of the proton (p) and the neutron-pion pair (iñ).

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admixture In the sense given earlier in these notes, for the purpose of describing elastic scattering we have approximately (compare eq (4.4))



p=p *+pTp. The production amplitude is:

Msi= (õ | T |P) dip d2an+ (ññ | T | ññ dp djan. Suppose the target is a nucleon, then (ñ |T|p) is taken to be the proton-nucleon elastic scattering amplitude, and (ññ | T | ññ) is constructed (e.g., à la Glauber) from (ñ | T | #) and (ñ | T | ñ) pion-nucleon and neutron-nucleon elastic scattering amplitudes [29].

Suppose the target is a nucleus: everything goes the same way except that elastic scattering amplitudes are taken to be with the nucleus (not a nucleon as before). Again they can be computed à la Glauber [29], [S4].

The standard noncovariant perturbation theory is an "obvious" tool to construct such states. Let H' be the interaction Hamiltonian which couples the states p) and ñi). Let this coupling be weak so that it is sufficient to take only the lowest order corrections:

(kpkr | H' | kp) | Vi(kp) )=2 || kp, 0, 0)+ E

| 0, kr, kn)+... Exp-Ern



+..] +...


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(pp | H' | pnpr)

| Pp, 0, 0)+... Epp-En-EPT



where Z is the renormalization constant,

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| kp, 0,0)= 1 p) is a state of a bare proton

0, 1, k = ( 7 ) is a state of a bare neutron-pion pair. Note that the sign of the second term in / V;) is "minus” because the order of energies in the denominator was changed. These two states should be orthogonal. Indeed

(Pnp. | H' | kp) (kp | H' | pnpr)* (Y; Y;)=2

=0 (Ek- EP - EP Ek-En-Ep because H' is hermitian, hence (kp | H' | Prpr)* = (prp. H' | kp). Note also that if H' is small, Z=1 to first order. So, we have:

(pnpr | H'| kp) diP=d,an=1 and d2P = - (d2an) * =


Exp-Epn - Epp Graphically, do corresponds to the vertex:




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